What's the opposite of primitive?

The most general term (language agnostic) is a composite data type.

In Java there are several types of composite data type, the most obvious is a Class but an Enum would be another example.

A boxed type is another related term, which is a primitive wrapped in a class to provide an object oriented interface. Examples of this in Java are an int vs. an Integer or a char vs. a Character.

In java, int, byte and boolean are primitives, while String is not.

A String is actually a boxed type. Under the hood it's implemented as a char array just like C and you could use a char array too if you wanted. Obviously most people don't do this as you won't have any of the convenient abstractions provided by the String class.

Here's how the java.lang.String class is implemented in openjdk:

/** The value is used for character storage. */
private final char value[];

As you can see, there's nothing special about a string in Java.